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\({\cal U}{\cal V}\)-theory of a Class of Semidefinite Programming and Its Applications

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Abstract

In this paper we study optimization problems involving convex nonlinear semidefinite programming (CSDP). Here we convert CSDP into eigenvalue problem by exact penalty function, and apply the \({\cal U}\)-Lagrangian theory to the function of the largest eigenvalues, with matrix-convex valued mappings. We give the first-and second-order derivatives of \({\cal U}\)-Lagrangian in the space of decision variables Rm when transversality condition holds. Moreover, an algorithm frame with superlinear convergence is presented. Finally, we give one application: bilinear matrix inequality (BMI) optimization; meanwhile, list their \({\cal U}{\cal V}\) decomposition results.

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Acknowledgements

We would like to thank the referees for their constructive criticism that helped to improve the presentation.

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Authors and Affiliations

  1. School of Science, Dalian Maritime University, Dalian, 116026, China

    Ming Huang & Jin-long Yuan

  2. School of Mathematical Science, Dalian University of Technology, Dalian, 116023, China

    Li-ping Pang & Zun-quan Xia

Authors
  1. Ming Huang
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  2. Jin-long Yuan
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  3. Li-ping Pang
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  4. Zun-quan Xia
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Corresponding author

Correspondence to Ming Huang.

Additional information

This paper is supported by the National Natural Science Foundation of China (Nos. 11701063, 11901075); the Project funded by China Postdoctoral Science Foundation (Nos. 2019M651091, 2019M661073); the Fundamental Research Funds for the Central Universities (Nos. 3132021193, 3132021199); the Natural Science Foundation of Liaoning Province in China (Doctoral Startup Foundation of Liaoning Province in China (Nos. 2020-BS-074); Dalian Youth Science and Technology Star (No.2020RQ047); Huzhou Science and Technology Plan (No.2016GY03); Key Research and Development Projects of Shandong Province (No. 2019GGX104089); and the Natural Science Foundation of Shandong Province (No. ZR2019BA014).

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Huang, M., Yuan, Jl., Pang, Lp. et al. \({\cal U}{\cal V}\)-theory of a Class of Semidefinite Programming and Its Applications. Acta Math. Appl. Sin. Engl. Ser. 37, 717–737 (2021). https://doi.org/10.1007/s10255-021-1037-5

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  • DOI: https://doi.org/10.1007/s10255-021-1037-5

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